Allan kindly replied to an issue of using interactions where both
variables are continuous,
I asked, "Do cont*cont interactions make sense?" and David Moore and
Constantine Daskalakis responded (quoted below). With respect, I think
both of them miss the distinction between having a mathematical
function
and having a model that usefully describes and helps intepret a
particular
dataset.
David Airey <[email protected]> provided this example looking
at
eye weight in mice:
. * full model
. regress eye body* age* brain* parity* litter* sex
I [RAR] assume that:
sex is categorical;
litter [size] is discrete over a small range of values, so could be
treated as ordered categorical;
parity is place in litter [?] so could be similar;
body [length? mass?], brain [mass] and age could all be continuous.
Yes, "sex" is a 1 0 dummy that would have been better named "female".
It could be argued that both litter size (1 to 15) and parity (1 to 11;
the number of litters born by the mother of the mouse in question) can
be treated as a continuous variable in that a linear or nonlinear
function might be expected. Inexperienced mothers may care less well
for their young. Larger litters might run into not getting enough care
from the mother. Indeed this seems to be the case. Parity and age are
confounded in this data set. Body is weight. Brain is weight.
On that interpretation, we expect eye, body and brain sizes all to be
positively correlated,
They are.
and various models involving body (or body^3 if
it's length), brain, body*brain, body^2, brain^2 might give similar
goodness of fit.
Scatters suggest power terms are not very important.
Size is correlated with age during the period of growth,
and there may be an argument that a young*(big for age) mouse has the
similar eye to a mature*(average size) mouse, so fitting body*age may
work
mathematically.
Surprisingly, within the adult mouse, eyes continue to grow with body
size in some strains. Some rodents exhibit almost indeterminate growth
patterns (not like fish of course but different than us for sure
(except in the US where most adults continue to grow side ways). Even
the brain in some strains keeps up to some extent, although not likely
from neurogenesis. The relationship between body and brain may show
different allometry (differential scaling of parts due to different
growth patterns) in different strains. This is a common observation
across species, but likely occurs within strains of mice to some extent
as well.
I think I get your point; biology and prior evidence could guide the
choices better. So maybe your point was that you prefer
evidence/thought based modeling and are uncomfortable with exploratory
kitchen sink approaches. Below you make the further point that treating
a cont*cont interaction by allowing one variable to be categorical
provides for easier interpretation, and you seem to have retracted the
earlier stronger statement somewhat.
Sometimes I get lazy and want a quick test of nonadditivity. I throw in
interactions en masse (e.g., all two way) and test the group of terms.
Other times I think about what I'm doing. Here the purpose was to
consider sources of variance that account for eye weight other than
specific genotypes which I've not discussed. I was not greatly
concerned about getting the best model, just removing obvious and
larger sources of variance from the error term such that I might have
better luck at detecting additive genotypes that explain eye weight.
That in itself might not make good sense, so I'll have to think about
all this a little more.
But how much easier to interpret if one of the continuous variables is
divided into classes, so the interaction coefficient now expresses the
difference in slope of the main regression according to group
membership.
The results now become plottable as a scatterplot with multiple lines.
I'm all for an easier interpretation.
Noting the proviso above about "correlated during the period of
growth",
one might look for periods of rapid growth and compare growth rates for
the eye and body. Or one could look at a multi-stage model (SEM),
postulating that body size is a function of age, and then relate eye
size
to body etc.
Over my head.
When I referred to an "act of faith", this was not to recommend a naive
metaphysical trust but rather to draw attention to the need to examine
the
basis and usefulness of a model that somehow "represents" the science.
As
I wrote, a computer will always do the arithmetic, whether or not the
sum
is justified.
Thanks, Allan, for the comments.
-Dave
> At 01:02 PM 6/26/03, David E Moore wrote:
> >Let me say at the outset that I applaud any conscientious effort to
> >examine the plausibility of nonlinear and/or nonadditive models.
> >... If it's always a leap of faith to have a linear additive
> >model, then perhaps one might want to think twice about creating
> >interaction terms that assume linear relationships.
On Thu, 26 Jun 2003, Constantine Daskalakis wrote:
> I think the question has things backwards. The point of the
interactions is
> that they allow us to relax model assumptions (notably, that of
> additivity). As such, they indeed make much sense -- they actually
allow us
> to fit a more flexible model.
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